Of all eligible voters in your county, 70% are currently registered to vote. A recent poll predicts that the relationship between V, the percentage of all eligible voters who are registered to vote, and t, the number of years from now will be modeled by the following equation. V=100-30*e^(-0.04 t) In how many years will 80% of all eligible voters in your county be registered to vote? Give an exact answer expressed as a natural logarithm. years

Question

Of all eligible voters in your county,  70% are currently registered to vote. A recent poll predicts that the relationship between  V, the percentage of all eligible voters who are registered to vote, and  t, the number of years from now will be modeled by the following equation.  V=100-30*e^(-0.04 t) In how many years will  80% of all eligible voters in your county be registered to vote? Give an exact answer expressed as a natural logarithm. years

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Set up equation with V: Set up the equation with V equal to 80. We are given the equation V = 100 - 30*e^(-0.04t) and we want to find the value of t when V is 80. So, we set V to 80 and solve for t. 80 = 100 - 30*e^(-0.04t)Subtract to isolate exponential term: Subtract 100 from both sides of the equation. 80 - 100 = 100 - 30*e^(-0.04t) - 100 -20 = -30*e^(-0.04t)Divide sides by -30: Divide both sides by -30 to isolate the exponential term. -20 / -30 = (-30*e^(-0.04t)) / -30 2/3 = e^(-0.04t)Take natural logarithm: Take the natural logarithm of both sides to solve for t. ln(2/3) = ln(e^(-0.04t))Simplify right side: Use the property of logarithms that ln(e^x) = x to simplify the right side of the equation. ln(2/3) = -0.04tDivide sides by -0.04: Divide both sides by -0.04 to solve for t. t = ln(2/3) / -0.04

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